# American Institute of Mathematical Sciences

January  2020, 16(1): 371-386. doi: 10.3934/jimo.2018157

## Analysis of strategic customer behavior in fuzzy queueing systems

 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author.Email address: math_zjc@csu.edu.cn (Jingchuan Zhang)

Received  June 2018 Published  January 2020 Early access  September 2018

Fund Project: This work is partially supported by the National Natural Science Foundation of China (11671404), and the Fundamental Research Funds for the Central Universities of Central South University (2017zzts061, 2017zzts386).

This paper analyzes the optimal and equilibrium strategies in fuzzy Markovian queues where the system parameters are all fuzzy numbers. In this work, tools from both fuzzy logic and queuing theory have been used to investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. By Zadeh's extension principle and $α$-cut approach, we formulate a pair of parametric nonlinear programs to describe the family of crisp strategy. Then the membership functions of the strategies in single and multi-server models are derived. Furthermore, the grated mean integration method is applied to find estimate of the equilibrium strategy in the fuzzy sense. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and a sensitivity analysis is performed, which show the relationship of these strategies and social benefits. Our finding reveals that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the system managers could get more precise information.

Citation: Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
##### References:

show all references

##### References:
The approximate membership function of optimal and equilibrium threshold $n$.
The approximate membership function of optimal and equilibrium arrival rate $\lambda$.
The approximate membership function of optimal social benefit $S$.
Fuzzy trapezoidal value for the input parameters $\tilde{\Lambda}$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
Fuzzy trapezoidal value for the input parameters $\tilde{\mu}$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{R}$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{C}$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
 [1] Wai-Ki Ching, Tang Li, Sin-Man Choi, Issic K. C. Leung. A tandem queueing system with applications to pricing strategy. Journal of Industrial & Management Optimization, 2009, 5 (1) : 103-114. doi: 10.3934/jimo.2009.5.103 [2] Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial & Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581 [3] Yahia Zare Mehrjerdi. A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021019 [4] Jinyan Wang, Yanni Xiao, Robert A. Cheke. Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5849-5870. doi: 10.3934/dcdsb.2019109 [5] Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036 [6] Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial & Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73 [7] Wenjuan Jia, Yingjie Deng, Chenyang Xin, Xiaodong Liu, Witold Pedrycz. A classification algorithm with Linear Discriminant Analysis and Axiomatic Fuzzy Sets. Mathematical Foundations of Computing, 2019, 2 (1) : 73-81. doi: 10.3934/mfc.2019006 [8] Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1999-2027. doi: 10.3934/jimo.2019040 [9] Qi Wang, Yue Zhou. Sets of zero-difference balanced functions and their applications. Advances in Mathematics of Communications, 2014, 8 (1) : 83-101. doi: 10.3934/amc.2014.8.83 [10] Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial & Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299 [11] Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 [12] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [13] Henk van Tilborg, Josef Pieprzyk, Ron Steinfeld, Huaxiong Wang. New constructions of anonymous membership broadcasting schemes. Advances in Mathematics of Communications, 2007, 1 (1) : 29-44. doi: 10.3934/amc.2007.1.29 [14] Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 [15] Xuemei Li, Rafael de la Llave. Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623 [16] Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353 [17] Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217 [18] Andreas Widder. On the usefulness of set-membership estimation in the epidemiology of infectious diseases. Mathematical Biosciences & Engineering, 2018, 15 (1) : 141-152. doi: 10.3934/mbe.2018006 [19] Xi-Nan Ma, Jiang Ye, Yun-Hua Ye. Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 225-243. doi: 10.3934/cpaa.2011.10.225 [20] Vikram Krishnamurthy, William Hoiles. Information diffusion in social sensing. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 365-411. doi: 10.3934/naco.2016017

2020 Impact Factor: 1.801